Spatial and temporal variability of the solar radiation heat flux in streams of a forested catchment

Abstract Solar radiation is generally the largest contributing flux to the heat budget of streams and its estimation is crucial to predict stream water temperature with process-based models. The objective of this research is to quantify the spatial (between-site comparison of different stream sizes, within-site comparison at the reach scale) and temporal (seasonal, daily and hourly scales) variability in the transmission coefficient, which represents the proportion of incoming solar radiation reaching streams. We measured solar radiation at an open site with a meteorological station and at microclimate sites located in three streams of various sizes in the Miramichi River basin (Canada). During the summer, the percentage of incoming daily solar radiation reaching a stream varied from 8% in a small headwater stream (Trib) to 43% in a medium-sized stream (CatBk) and was close to 100% in a wide river (LSWM). We observed the largest variability between transmission coefficients for different stream sizes (range of variation = 92%) due to very different canopy closures, followed by variability at the reach scale between lateral positions (range = 21% between left and right banks) and between longitudinal positions (range = 11% between upstream and downstream sites), as measured at the medium-sized stream. Temporal variability was greatest at the seasonal scale where the transmission coefficient varied by 23% between May and September at the small headwater stream. The hourly variability of the transmission coefficient (i.e. associated with different solar angles) surpassed daily variability (i.e. associated with different cloud cover conditions), with coefficients of variation computed at the hourly time scale three to five times greater than at the daily time scale. Overall, this research offers insight regarding the handling of spatial and temporal variability of solar radiation which should provide further insight to improve process-based stream temperature models.

Le rayonnement solaire est g en eralement le plus grand flux contribuant au bilan thermique des cours d'eau et son estimation est cruciale pour pr edire la temp erature de l'eau a l'aide de mod eles d eterministes. L'objectif de cette recherche est de quantifier la variabilit e spatiale (comparaison entre des cours d'eau de diff erentes tailles et a l' echelle du tronçon au sein d'un même cours d'eau) et temporelle ( echelles saisonni ere, journali ere et horaire) du coefficient de transmission, qui repr esente la proportion de rayonnement solaire entrant atteignant un cours d'eau. Nous avons mesur e le rayonnement solaire a un site ouvert avec une station m et eorologique et a des sites microclimatiques situ es dans trois cours d'eau de diff erentes tailles dans le bassin de la rivi ere Miramichi (Canada). Au cours de l' et e, le pourcentage de rayonnement solaire entrant journalier atteignant un cours d'eau variait de 8 % dans un petit cours d'eau de tête (Trib) a 43 % dans un ruisseau de taille moyenne (CatBk) et etait proche de 100 % dans une large rivi ere (LSWM). Nous avons observ e la plus grande variabilit e entre les coefficients de transmission de cours d'eau de diff erentes tailles ( ecart de variation ¼ 92 %) en raison de fermetures de canop ee tr es diff erentes, suivie d'une variabilit e a l' echelle du tronc¸on entre diff erentes positions lat erales ( ecart ¼ 21 % entre les rives gauche et droite) et entre diff erentes positions longitudinales ( ecart ¼ 11 % entre les sites amont et aval), telles que mesur ees dans le ruisseau de taille moyenne. La variabilit e temporelle etait la plus grande a l' echelle saisonni ere o u le coefficient de transmission variait de 23 % entre mai et septembre dans le cours d'eau de tête. La variabilit e horaire du coefficient de transmission (c'est-a-dire associ ee a diff erents angles solaires) d epassait la variabilit e quotidienne (c'est-a-dire associ ee a diff erentes conditions de couverture nuageuse), avec des coefficients de variation calcul es au pas de temps horaire trois a cinq fois plus elev es qu'au pas de temps journalier. Dans l'ensemble, cette recherche permet de mieux prendre en compte la variabilit e spatiale et temporelle du rayonnement solaire, ce qui devrait guider les efforts de mod elisation de la temp erature des cours d'eau.

Introduction
Water temperature plays an important ecological role in freshwater ecosystems and changes in stream temperatures can significantly impact fish distribution, growth, mortality, habitat use and community dynamics (Caissie 2006). In order to address these important fish habitat issues, different statistical and processbased (deterministic) modelling approaches have been developed to predict stream water temperatures (Caissie, Satish, and El-Jabi 2005;Benyahya et al. 2007;Leach and Moore 2017). Statistical and processbased models differ in complexity and data requirements. For instance, process-based models are efficient tools to understand physical processes governing the stream heat budget, but require numerous input data, including hard-to-obtain above-stream microclimate data. Such models have been used to predict the impact of various anthropogenic disturbances to the thermal regime of streams (Lowney 2000;Wu et al. 2012;Dugdale, Curry, et al. 2018) and assess different management scenarios (Cristea and Burges 2010;Garner et al. 2017;Null, Mouzon, and Elmore 2017). There are also arguments in favor of process-based models for climate change studies given the inability of statistical models to accurately project water temperatures over time in certain regions, especially for conditions falling outside of those used for model calibration (Arismendi et al. 2014). Statistical models remain useful when limited hydrometeorological data are available and as such, they are a particularly powerful tool to model stream water temperature at large spatial scales (Isaak et al. 2014).
The use of process-based models requires a good understanding of physical processes governing heat exchange between the stream and its environment. Heat fluxes need to be quantified both at the streamair interface as well as at the stream-streambed interface. Among the different fluxes at the stream-air interface, shortwave radiation (i.e. incoming solar radiation) is generally the largest contributing flux to the heat budget of streams (Webb and Zhang 1997;Johnson 2004;Hebert et al. 2011). A proper estimation of the solar radiation heat flux is therefore critical in the application of process-based water temperature models. In forest environments, stream shading by riparian vegetation highly influences the amount of incoming solar radiation reaching the stream and therefore its contribution to the heat budget. For example, solar radiation can represent between 15 and 89% of summer heat gains in forested catchments and the contribution of this heat flux varies according to the type of canopy as well as the density of the riparian vegetation (Webb and Zhang 2004;Hannah et al. 2008;Hebert et al. 2011;Simmons et al. 2015). Many studies have shown that the removal of riparian vegetation, through forestry activities or wildfires, increases the solar radiation heat flux to streams which leads to increases in stream temperatures (Moore, Spittlehouse, and Story 2005;Wagner et al. 2014). In forested environments, solar radiation reaching a stream is also governed by the position of the sun relative to the streamside vegetation. As such, its estimation is a complex task that needs to take into account the daily and seasonal movement of the sun, daily changes in cloud cover as well as the seasonally varying influence of riparian vegetation, especially in deciduous forests. Moreover, solar radiation reaching a stream is a combination of direct (beam radiation traveling in a straight line from the sun to the earth's surface), diffuse (scattered by the atmosphere and emitted uniformly from the sky) and reflected (reflected by terrestrial objects) radiation. The reflected solar radiation component at the stream-air water interface is generally small and in the range of 3% to 5% (Sinokrot and Stefan 1993;Caissie, Satish, and El-Jabi 2007). Riparian vegetation can influence these components differently, which adds another layer of complexity in the estimation of this flux. Solar radiation reaching streams in forested catchments has generally been estimated with two different approaches, namely through direct measurements of solar radiation above and under the forest canopy (Brosofske et al. 1997;Benyahya et al. 2010) or by inferring solar radiation reaching the stream under the canopy based on above-canopy solar radiation and riparian vegetation characteristics (Leach and Moore 2010;Bulliner and Hubbart 2013;Moore, Leach, and Knudson 2014). In the direct measurement approach, the proportion of incoming solar radiation reaching the stream is assessed by comparing under-canopy conditions to above-canopy or open-site measurements. This approach has been extensively used to assess the attenuation of solar radiation by the canopy in closed forest environments (Reifsnyder, Furnival, and Horowitz 1971;Vales and Bunnell 1988), and has also been used to quantify the solar radiation heat flux reaching streams (Caissie, Satish, and El-Jabi 2007;Dugdale, Malcolm, et al. 2018). The average ratio of incoming solar radiation reaching the stream, or conversely, the 'shade factor' describing the proportion of incoming solar radiation blocked by riparian vegetation, can afterwards be used in process-based stream temperature models (Dugdale, Hannah, and Malcolm 2017). The advantage of the direct measurement approach is that it provides an estimation of all types of radiation (direct, diffuse, reflected) reaching the stream. However, the deployment of sensors in streams can be problematic, especially during high flow events, which can damage or destroy equipment. Moreover, capturing the spatial variability of solar radiation remains a challenge with this approach given the need for numerous point measurements, especially as the canopy closure become more important (e.g. mediumsized and small streams).
The riparian characterization approach estimates solar radiation reaching the stream based on characteristics of the riparian vegetation estimated from field observations (e.g. leaf area index, spherical densitometer, hemispherical photographs) or geographic information systems (e.g. LiDAR, vegetation inventory data). This approach typically requires partitioning incoming solar radiation into direct and diffuse components using empirical models (Erbs, Klein, andDuffie 1982 in Leach andMoore 2010;Bulliner and Hubbart 2013;Sun et al. 2015, Vignola and McDaniels 1984in Chen et al. 1998. Notably, both direct and diffuse components can be measured in open sites using a combination of pyranometers and sun-shading ring pyranometers. Once the partitioning is done, the direct solar radiation reaching the stream is modelled based on the solar position (azimuth and elevation), the stream orientation and the shading effect of topography, stream banks and riparian vegetation (Li, Jackson, and Kraseski 2012). Hemispherical photographs have often been used in field studies to characterize the above-stream opening in the canopy and, based on the sun's position at a given moment of the day, one can determine if the direct beam radiation is intercepted by vegetation or passing through (Ringold et al. 2003;Leach and Moore 2010;Bulliner and Hubbart 2013). Diffuse solar radiation reaching the stream depends on the sky-view factor (i.e. portion of the sky dome visible looking upward from the stream surface) which can be estimated from hemispherical photographs (Leach and Moore 2010) as well as from geometric calculations (Moore, Leach, and Knudson 2014). An approach based on the characterization of the riparian vegetation offers the possibility of capturing the spatial variability of reach-scale controls of the solar radiation heat flux. However, finding detailed data to describe riparian vegetation at the regional scale is generally difficult, although LiDAR data may offer a way forward (Wawrzyniak et al. 2017;Loicq et al. 2018). Moreover, the partitioning of solar radiation into direct and diffuse components remains an estimated value derived from calculations and field validation remains sparse in the literature. Single-point measurements of above-steam solar radiation may not be representative of the reach-scale averaged solar radiation, although they provide a good estimate of solar radiation, as experienced by the stream, at specific points. For example, Leach and Moore (2010) measured solar radiation variability at the reach level, and found that single location measurements did not represent well the modelled average reach conditions. While field methods using photographs (Li, Jackson, and Kraseski 2012) or a canopy analyzer (Davies-Colley and Payne 1998) have been used to quantify shade in streams, they have rarely been implemented to validate the estimation of solar radiation reaching streams using riparian characterization approaches.
It is clear from the literature that estimating solar radiation reaching streams in forested environments remains a challenge. Both direct measurement and riparian characterization approaches play a key role in the overall understanding and validating of this important heat flux for stream temperature models. In the present study, we used a direct measurement approach to assess solar radiation reaching streams of various sizes in forested catchments. These data should provide important information to the overall understanding of solar radiation reaching streams. In fact, very few studies have measured solar radiation reaching streams of different sizes within the same meteorological setting. The objective of the present study is therefore to quantify the spatial (between-site comparison of different stream sizes, within-site comparison at the reach scale) and temporal (seasonal, daily and hourly scales) variability in the proportion of incoming solar radiation reaching a stream as derived with the direct measurement approach. Some key hypotheses of this study are that: i. In terms of spatial variability, transmission coefficients will increase with stream size, with a low proportion of solar radiation reaching small headwater streams and a large proportion reaching large and wide rivers. Transmission coefficients of medium-sized streams will exhibit considerable spatial variability given lateral variations in the shading influence of riparian vegetation, particularly for direct solar radiation, and given longitudinal variations in the stream azimuth.
ii. In terms of temporal variability, transmission coefficients will show the most variation at the hourly time scale as a result of gaps in the riparian canopy which will allow direct radiation to reach the stream at specific angles. Day-to-day variability in transmission is expected as a result of changing cloud cover conditions, but to a lesser level than at the hourly time scale.

Study area
This study was conducted within the Miramichi River basin in New Brunswick, Canada ( Figure 1). The region is characterized by a humid climate with a mean annual precipitation averaging 1072 mm (Environment Canada, Canadian Climate Normal, 1981 and annual evapotranspiration estimated at 416 mm (data 1969-1991, Caissie and El-Jabi 1995. July is the warmest month of the year with a monthly mean air temperature of 19.1 C (Environment Canada, Canadian Climate Normal, 1981. Vegetation is composed of conifers (65%, mainly balsam fir) and deciduous trees (35%, mainly sugar maple, red maple, white birch, yellow birch and trembling aspen) (Cunjak, Caissie, and El-Jabi 1990). Riparian vegetation reached an average height of 14 m, as assessed using a digital surface model (Natural Resources Canada, High Resolution Digital Elevation Model, CanElevation series). Within the Miramichi River basin, we measured solar radiation at three sites along the stream continuum: a small headwater stream (Tributary 1, Trib), a medium-sized stream (Catamaran Brook; CatBk) and a large and wide river (Little Southwest Miramichi River; LSWM, Figure 1). The first site, Trib, has a drainage area of 4.5 km 2 and a mean wetted width of 1.7 m. This site is the most sheltered with a canopy closure estimated visually at about 85%. The second site, CatBk, flows into LSWM and has a drainage area of 28.7 km 2 at the study site. CatBk is relatively well sheltered by riparian vegetation along the stream with a canopy closure estimated visually to vary between 45 and 60%. Given the longitudinal variation in canopy closure, one site was monitored in 2012 at CatBk and two sites were monitored in 2014 (CatBk1 and CatBk2). In 2012, the canopy closure at CatBk was not measured but was estimated at a typical value of 50%. In 2014, the most upstream site (CatBk1) was more sheltered with a canopy closure estimated at 55% and a width of 6.6 m. The downstream site (CatBk2) was slightly more open with a canopy closure of about 40% and a width of 6.5 m.
The third site, LSWM has a drainage area of 1190 km 2 and a mean wetted width of approximately 80 m. The canopy closure at LSWM was estimated at less than 20%. Solar radiation measurements at the three microclimate sites were compared with open-site solar radiation measurements at a meteorological station (Met). The meteorological station is located in the CatBk basin at less than 7 km from microclimate sites in an open area where trees have been cut (approximately 200 m Â 200 m; Figure 1).

Methods
This study integrates solar radiation measurements taken between 2007 and 2014. We measured solar radiation at the open site Met from 2007 to 2014 with a silicon pyranometer at about 2 m above the ground. At the three microclimate sites (Trib, CatBk, LSWM), we measured solar radiation also using silicon pyranometers (Kipp & Zonen SP LITE2 and LI-COR) at 2 m above the stream surface. At Trib, we measured solar radiation in 2007, 2008, 2012, 2013 and 2014. We collected year around data at this station as it was a permanent site. At LSWM, we measured solar radiation during the summers of 2007, 2008 and 2012 as this station was removed due to late autumn high flow conditions and spring ice conditions (often associated with ice jams). At Trib and LSWM, we installed a microclimate station within the channel to measure mid-stream values of solar radiation. At CatBk, we measured solar radiation at one site in the summer of 2012 (CatBk) and two sites in the summer of 2014 (CatBk1, CatBk2). In 2012, we installed a sensor at approximately 25% of the stream width on the right bank of CatBk. In 2014, we installed four sensors laterally across the stream at approximately equal distance among each other for both sites, as shown in Figure 2. In the present study, we compared solar radiation at microclimate sites (Trib, CatBk or LSWM) with simultaneous measurements of solar radiation at the open site (Met). We computed the solar radiation transmission coefficient (Campbell and Norman 1998) as the average ratio between solar radiations measured at the microclimate and open sites: where s is the transmission coefficient, n is the number of days or hours considered, K i, stream is the abovestream solar radiation measured at time step i and K i, open is the solar radiation measured at the open site at time step i. We computed the transmission coefficient for the summer period (July-August). We computed this coefficient at daily and hourly time scales and, for the hourly time scale, we only considered daylight hours (between 7:00 and 19:00, inclusively).
As an example, a transmission coefficient of 90% indicates that 90% of the incoming solar radiation reaches the stream and 10% is intercepted by the vegetation. For CatBk, we pooled together the solar radiation measurements at the eight sensors deployed in 2014 (four at CatBk1 and four at CatBk2) and used the average to compute the transmission coefficient. When comparing the present method to other studies, the transmission coefficient computed in the present study is equivalent to one minus the shading factor used in other studies such as Sinokrot and Stefan (1993). While our study focuses on global radiation (sum of direct, diffuse and reflected radiation) reaching steams, other studies have computed the proportion of incoming solar radiation reaching a stream only on days with completely diffuse conditions, an important component to guide the restoration of riparian vegetation (Rutherford, Davies-Colley, and Meleason 2018). It should also be noted that streams with similar sky-view factors could vary in terms of the proportion of diffuse solar radiation reaching the stream depending on how efficiently solar radiation is transmitted through the canopy (e.g. foliage transmittance and reflectivity). Regardless of similarities in sky-view factors, the proportion of direct solar radiation reaching a stream is expected to vary as a function of the sun's position relative to riparian vegetation.
As mentioned above, we used silicon pyranometers to measure below canopy (above-stream) solar radiation at microclimate sites. These photoelectric sensors are less expensive than thermoelectric sensors (e.g. Eppley) sensors but only sample a portion of the full shortwave spectrum (400 to 1100 nm). Moreover, silicon sensors are calibrated under direct sunlight conditions and below-canopy radiation has different spectral properties then incoming radiation, with photosynthetically active radiation (PAR) generally depleted under the canopy given their preferential absorption by leaves (Baldocchi et al. 1984). Consequently, silicon pyranometers provide an incomplete, but still informative assessment of solar radiation. In this study, we considered differences in above-stream and open-site solar radiation magnitude but part of our analysis looked into transmission coefficients. By focusing on ratio values, we can minimize limitations associated with the fact that silicon pyranometers only measure a portion of the shortwave spectrum. Moreover, it would have been cost prohibitive to deploy numerous thermoelectric sensors and while there are certain issues associated with the spectral response of silicon pyranometers, they remain the most practical and a widely used instrument to answer questions relative to the spatial variability of solar radiation.
We assessed the influence of sunny (combination of direct and diffuse radiation) vs. overcast (diffuse radiation only) conditions on the transmission coefficient by computing the daily sky clearness ratio (Leach and Moore 2010): where C is the sky clearness ratio, K incident is the incoming solar radiation measured at the open site (W m À2 ) and K max is the maximum incoming solar radiation under a cloud-free sky (W m À2 ) which we computed as: where W is the clear sky transmissivity (0.75; Leach and Moore 2010) and K global is the global horizontal solar radiation (W m À2 ) which we computed using the R package solaR (Perpiñ an 2012).

Between-site comparison of transmission coefficient for different stream sizes
The proportion of solar radiation reaching the stream increased with the stream size (Table 1, Figure 3). At the daily time scale, 8% of incoming solar radiation reached Trib, 43% reached CatBk and 101% reached  (daily) and 39% (hourly), although it should be reminded that transmission coefficients at CatBk were computed from an average of eight sensors, which inherently reduced variability. The coefficient of variation of transmission coefficients also increased about three to fivefold between the daily and hourly scales. Figure 3 clearly illustrates the increased variance at the hourly scale for all sites. The more sheltered sites (Trib and CatBk) showed more scatter at the hourly time scale, especially at CatBk when solar radiation at Met was greater than 600 W m À2 . At CatBk, this scatter was associated with large above-stream solar radiation (> 400 W m À2 ) recorded mid-afternoon between hours 14 and 16 ( Figure S1, Supplementary material). At Trib, scatter is associated with large solar radiation values above the stream early in the morning between hours 7 and 9 ( Figure S1).

Within-site comparison of transmission coefficients at the reach scale
Reach-scale variability of transmission coefficients was only investigated at the medium-sized stream (CatBk) where eight pyranometers were deployed at various lateral (n ¼ 4, left to right bank) and longitudinal (n ¼ 2, upstream CatBk1 and downstream CatBk2) positions. At the daily time scale, the transmission coefficients varied by 21% (46% vs. 25%) from the left to the right bank at CatBk1 and by 9% (52% vs. 43%) at CatBk2 (Table 2). At CatBk1, the left site showed a daily transmission coefficient close to 45% compared to values of 25% and 36% at the right side, with an overall average of 38% (standard deviation ¼ 6%). At CatBk2, the average transmission coefficient was greater than at CatBk1, reaching 49% (standard deviation ¼ 4%). We observed less lateral variation in transmission coefficients with values close to 50% except on the far left where the transmission coefficient was 43%. Figure 4a and 4b also show that the daily lateral variability was greater at the more sheltered site (CatBk1) than at the more open site (CatBk2).
Notably, the slopes of equations were very similar to the mean value of transmission coefficients in Table 2. Compared to CatBk2, we observed more scatter at CatBk1, especially for large values of solar radiation at the Met site (> 200 W m À2 ). Lateral variability of transmission coefficients was similar at the hourly scale, with a 16% difference (42% vs 26%) from the left to the right bank at CatBk1 and 10% (50% vs 40%) at CatBk2 (Table  2). Overall, mean coefficients at the hourly time scale were very similar to those at the daily time scale, although their variability increased. The coefficient of variation of the average transmission coefficient at CatBk1 and CatBk2 varied between 8% and 15% at the daily time scale compared to 38% to 47% at the hourly time scale (Table 2). When averaging lateral measurements at each site, considerable longitudinal differences persisted in transmission coefficients, with a 11% difference between CatBk1 (38%) and CatBk2 (49%) at the daily time scale (Table 2). Similar results were observed when comparing slopes at the reach scale (0.37 vs. 0.47; Figure 4c). Figure 5 shows how transmission coefficients varied hourly at each microclimate site during the summer. At the headwater stream Trib, we observed moderate hourly variations in the transmission coefficients ( Figure 5a). The median transmission coefficients varied between 6% and 9%, except early in the day (hours 7-8) when we observed larger values (median ¼ 15%, Figure 5a) during low solar angles. Unlike Trib that showed low hourly variance, the CatBk site showed high variability in transmission coefficients throughout the day (Figure 5b). The median transmission coefficients varied between 25% and 43% early in the day (hours 7-14), increased above 50% in the afternoon (hours ¼ 14-16) and then dropped below 40% in early evening. At LSWM, we observed small hourly variations in the transmission coefficients with a median value close to 1 throughout the day (Figure 5c). However, transmission coefficients were below 0.5 and above 2 on a few occasions, often early in the day (hours 7-8). Figure 6 shows hourly solar radiation measurements at the different study sites between day 186 (July 4) and 194 (July 12). As shown on this figure, solar radiation measured at LSWM and the Met site (open site) were of similar magnitude. However, small differences in solar radiation at LSWM and Met sites resulting from differences in cloud cover can translate into large variations in the transmission coefficient throughout the day (Figure 6b). For example, the ratio between solar radiation at LSWM and Met sites varied between 0.7 and 1.8 (average ¼ 1.1) on cloudy, overcast days (days 187-188), and was slightly lower varying between 0.3 and 1.5 (average ¼ 0.9) during sunny days (day 189-193). The transmission coefficient at CatBk was also variable during the day, generally low in the morning and then peaked towards the end of the day. Indeed, toward the end of the day, solar radiation at CatBk was close to solar radiation measured at the Met site. Trib showed relatively low solar radiation inputs and thus a small transmission coefficient. Overall, CatBk and Trib showed relatively stable transmission coefficients during overcast days (days 187-188) while we observed much more variability on sunny days (days 189-193; Figure 6b). Figure 7 shows the influence of overcast vs. sunny conditions on the daily transmission coefficient. At the two smaller streams (Trib and CatBk), the transmission coefficient decreased with sky clearness, meaning that a greater proportion of incoming solar radiation reached the stream during overcast days. For example, the transmission coefficients were close to 10% during overcast days (i.e., sky clearness ratio < 0.25) at Trib and decreased to 7% during sunny days (sky clearness ratio > 0.6; Figure 7a). At CatBk, transmission coefficients were about 50% during overcast days and decreased to about 40% on sunny days (Figure 7b). While the transmission coefficient was significantly correlated to the sky clearness ratio at the three sites (p < 0.05), the correlation was very weak at LSWM (R 2 ¼ 0.08, Figure 7c). For Trib and CatBk, we observed a stronger relationship between the sky clearness and transmission coefficients, with coefficients of determination (R 2 ) of 0.35 and 0.38 respectively. Table 3 shows the monthly variation of the mean daily transmission coefficient at Trib, the only site with data spanning the spring and summer periods. The daily transmission coefficients decreased from May (30%) to June (12%) and plateaued close to 7% in the summer (July to September). Slightly larger variations in the transmission coefficients were Figure 5. Hourly variation in transmission coefficients at (a) the headwater stream Trib, (b) the medium-sized stream CatBk and (c) the wide river LSWM. Orange diamonds represent the mean and in black, the central mark is the median, edges of the box represent interquartile range, whiskers represent extreme values that are not considered outliers and circles represent outliers. § Note that seven outliers with a transmission factor greater than 1.05 (min ¼ 1.17 and max ¼ 6.43) do not appear for hour ¼ 7 at Trib. observed in June (coefficient of variation ¼ 40.7%) as opposed to other months when the coefficient of variation was below 25% (Table 3). This large variation in the transmission coefficient in June coincides with the springtime leaf development of deciduous trees in the study region. The higher variability in September (33%) is most likely due to the falling of leaves which occurs during the latter part of this month.

Spatial variability of transmission coefficients in a forested catchment
We observed the largest variability between transmission coefficients at the catchment scale (range of variation ¼ 92%), followed by variability at the reach scale between lateral positions (range ¼ 21% between  left and right bank) and between longitudinal positions (range ¼ 11% between upstream and downstream site), as measured at CatBk. Between streams of different sizes, the daily average transmission coefficients varied from 8% at Trib to close to 100% at LSWM due to very different canopy closures (Table 1, Figure 3). Using a theoretical model of daily radiation at the stream level, DeWalle (2008) showed that the shading effect of riparian vegetation is greatest for streams with a vegetation height (H) to stream width (W) ratio greater than 1.5. Both Trib (H/W ¼ 8.2) and CatBk (H/W ¼ 2.1) exceeded this value and more than 50% of the incoming solar radiation was blocked by riparian vegetation at these sites. DeWalle (2008) showed that the shading effect of riparian vegetation is minimal for H/W smaller than 0.4. This is consistent with observations at LSWM which has a H/W of 0.2 and a transmission coefficient close to 100%. Reach-scale variability can also be important, as was observed in the medium-sized stream CatBk (Table 2, Figure 4). Laterally, the daily average transmission coefficients varied by as much as 21% between the left and right banks at CatBk1. Data on lateral variations of solar radiation reaching streams is sparse in the literature. Using models with geometries representing straight and meandering reaches, Rutherford, Davies-Colley, and Meleason (2018) measured diffuse light reaching the stream surface with a canopy analyzer and showed important lateral variations in stream shade, thus stressing the need to consider average solar radiation across the reach and not just mid-channel values. Similarly, Moore, Leach, and Knudson (2014) showed considerable variability in sky-view factors computed with a geometric model across a 10-m wide stream for relatively simple configurations in terms of riparian vegetation and bank geometry. Our results also indicate that the lateral position within streams strongly influences the magnitude of transmission coefficients in medium-sized streams ( Figure 4) and thus lateral variability needs to be considered when estimating the overall solar radiation for stream temperature modelling.
Longitudinally, the daily average transmission coefficient varied by 11% between CatBk1 (38%) and CatBk2 (49%, Table 2). This variation is likely the result of changes in riparian vegetation as CatBk1 had an estimated canopy closure of 55% compared to 40% at CatBk2. The variation in transmission coefficients could also be due to changes in the stream azimuth, as CatBk1 flows along a West-to-East axis while CatBk2 is oriented along a Southwest-to-Northeast axis at CatBk2 (Figure 1). Indeed, DeWalle (2008) showed that shade restoration on streams with an East-West azimuth such as CatBk1 requires taller riparian vegetation than streams with a North-South azimuth.
Given important longitudinal and lateral variations in the transmission coefficient measured at CatBk, we recommend that a careful assessment of spatial variability be performed when using a direct measurement approach in medium-sized streams (H/W % 2). In the present study, we deployed multiple fixed pyranometers to capture the spatial variability of transmission coefficients which allowed us to quantify the average transmission coefficient at the reach scale. While this approach required a large number of sensors and thus can be cost prohibitive for practical applications, simple and low-cost methods have also been developed to quantify the spatial variability of stream shade. For example, Davies-Colley and Payne (1998) proposed a method using canopy analyzers, one fixed at an open site and a second mobile sensor that takes equally spaced point measurements along a 100 m stream reach to quantify the average diffuse solar radiation reaching the surface of small streams (width ¼ 0.8 to 4 m). Li, Jackson, and Kraseski (2012) also developed a field method to measure stream shade by computing the number of grid points in the shade in upstream or downstream-pointing photographs. They used this photographic shade estimation technique to validate a model simulating stream shading by riparian vegetation.
We did not measure reach-scale variability of transmission coefficients at Trib and LSWM, as this would have required a large number of additional pyranometers. However, given the small H/W ratio (0.2) at LSWM, we would expect only minimal reach-scale variability, as mid-channel measurements of solar radiation were almost identical to those at Met (transmission coefficient close to 100%, Table 1). The deployment of sensors over a lateral transects would have been useful to better take into account the shading effect of riparian vegetation near the banks. However, most of the LSWM was not shaded during high solar angle periods (mid-morning to late afternoon), when solar radiation is at its peaked and makes its largest contribution to the heat budget of streams. At Trib, we do not expect lateral transects of above-stream solar radiation measurements to be very informative because of the small size of this stream and its almost complete canopy closure. However, longitudinal measurements would have been relevant to estimate the spatial variability of solar radiation reaching headwater streams. Overall, we would expect the transmission coefficient as well as its spatial variability to be very similar to what is measured in mixed forest stands. The daily average transmission coefficient at Trib (8%, Table 1) was similar to PAR transmittance values observed in mixed stands which range between 2% and 10% (Brown and Parker 1994;Canham et al. 1994;Messier, Parent, and Bergeron 1998). Even under a closed canopy, spatial variability of solar radiation transmission can be important. For example, the spatial coefficient of variation in daily PAR transmittance can exceed 40% in mixed forest stands Bergeron 1998). Still, Reifsnyder, Furnival, andHorowitz (1971) assessed sampling requirements to account for spatial variability when measuring below-canopy solar radiation and they estimated that one sensor was sufficient to estimate the average solar radiation in a deciduous stands while ten sensors would be needed in coniferous stands.

Hourly variability of transmission coefficients
Overall, there was a general agreement in the magnitude of transmission coefficients at hourly and daily time scales, with at most a 2% difference observed between these time scales (Table 1). This agreement between hourly and daily transmission coefficients has been used as a basis to reduce sampling requirements to estimate below-canopy light availability in forest stands. Indeed, Parent and Messier (1996) showed a strong correlation between instantaneous measurements and daily average values of below-canopy PAR. The largest solar radiation inputs to streams occurred between hours 10 and 17 ( Figure 6) and transmission coefficients at Trib and LSWM showed little variability during those hours ( Figure 5). For example, the greatest variability in transmission coefficients occurred in the early morning at Trib and LSWM (Figure 5a-c) when incoming solar radiation is generally low (e.g. summer average solar radiation at Met ¼ 24 W m À2 at hour 7 vs 596 W m À2 at hour 13). At the small headwater stream Trib, early-morning variability was likely the result of direct solar radiation coming through small gaps in the canopy in the early morning leading to large values of solar radiation measured above the stream (>100 W m À2 , Figure S1a). At the large and wide river LSWM, variability in transmission coefficient occurred throughout the day, with transmission coefficients greater than 2 measured early in the morning, midday or in the late afternoon ( Figure 5c). This variation in transmission coefficients was likely the result of differences in could cover at LSWM vs. at the open-site meteorological station, leading to direct solar radiation reaching the pyranometer at one location but not the other. Overall, this variability in transmission coefficients will likely not largely influence the heat budget computation in stream temperature models for small headwater streams (Trib) or large and wide rivers (LSWM).
On the other hand, we observed substantial variability in transmission coefficients at CatBk, with a large increase between hours 14 and 16 (Figure 5b), likely the result of a gap in the riparian canopy that allows direct solar radiation at that particular sun's position to go through and reach the stream surface. Indeed, the transmission coefficient increase from an average of 36% in the morning to an average of 61% between hours 14 and 16 ( Figure 5). This increase of 25% in the transmission coefficient coincides with peak solar radiation ( Figure 6) and thus would have an important impact on the computation of the heat budget of streams at the hourly time scale. For example, using an average transmission coefficient (41% , Table 1) to compute the daily average solar radiation reaching CatBk would lead to a 7% (12 W m À2 ) underestimation compared to using an hourly varying transmission coefficient ( Figure 5). While this variability should be considered when modeling stream temperatures at an hourly time scale, it will have little influence on stream temperature models over longer time scale (e.g. daily, weekly). Indeed, Table 1 shows very similar transmission coefficient values when averaged at a daily (43%) vs. an hourly (41%) time scale.

Daily variability of transmission coefficients and its relationship with sky clearness
As observed in the present as well as other studies, the variability of transmission coefficients decreased when averaged over longer periods (Battaglia et al. 2003;Comeau, Gendron, and Letchford 1998). Indeed, we observed a three to fivefold decrease in the coefficients of variation of transmission coefficients computed at the hourly vs. daily time scale (Table 1). Still, a certain level of variability persisted at the daily scale, which we found was correlated to sky clearness at the two smaller streams (CatBk, Trib) but not at the larger river (LSWM, Figure 7). Transmission of solar radiation increased with the cloud cover at Trib and CatBk, although solar radiation reaching the stream was overall of lower magnitude during overcast days. In other words, solar radiation transmission was lower on clear sunny days than on cloudy overcast days. When gaps in the canopy intersect little with the solar path and allow mainly for the transmission of diffuse rather than direct radiation, we can indeed expect the transmission coefficient to be lower on clear sunny days given a large proportion of direct solar radiation is blocked by the canopy. Vales and Bunnell (1988) observed a similar decrease in the transmission of global radiation during clear sunny days thus suggesting the need to consider direct and diffuse components separately when modelling solar radiation below canopies. For stream temperature modeling purposes, relationships such as those obtained in Figure 7 could be used to take into account such variability. This appears particularly important for small headwater streams such as Trib where the transmission coefficient can be reduced by half between overcast and sunny days (Figure 7a), even though, in terms of magnitude, solar radiation reaching the stream remains small in general.

Seasonal variability of transmission coefficients in a headwater stream (Trib)
At Trib, temporal variability of the transmission coefficient was greatest at the seasonal scale, going from a monthly average of 30% in May to a monthly average of 7% at the end of the summer (Table 3). A similar seasonal reduction in transmission has been reported for mixed forest stands, going from 22% of PAR transmitted below the canopy in the spring to 6% in the summer (Constabel and Lieffers 1996). These results stress the importance of accounting for seasonal differences in solar radiation transmission when modelling stream temperatures. This is especially true in headwater streams where we can expect the largest seasonal variation in transmission coefficients given less than 10% of the incoming radiation reaches the stream in the summer when the canopy becomes fully developed. For this reason, any streamside vegetation removal will have a significant impact on the overall stream budget and corresponding stream temperatures (Moore, Spittlehouse, and Story 2005).

Implications for process-based stream temperature modelling
At both daily and hourly time scales, we found a good fit between solar radiation measured at the open site and measured above LSMW, with a coefficient of correlation (r) greater than 0.96 (Figure 3e-f). For wide and large rivers, solar radiation measurements at a nearby site appear suitable to simulate the solar radiation heat flux in a process-based stream temperature model. The main issue to consider is the proximity between the river and solar radiation measurements, especially for sunny days as small differences in cloud cover can lead to differences at the hourly time scale in direct solar radiation reaching the river and the open site. At a daily time scale, we found a good fit between solar radiation measured at the open site and above Trib and CatBk, with r greater than 0.80 ( Figure 3). In small and mediumsized streams, using a single transmission coefficient to simulate solar radiation appears like a sound approach when implementing process-based stream temperature models at a daily time step during the summer period when little changes occur in canopy cover. At the hourly time scale, using a similar approach may yield more uncertainty as we observed more variability between solar radiation measured at the open site and above Trib (r ¼ 0.79) and CatBk (r ¼ 0.55). Capturing the temporal and spatial variability in solar radiation reaching the medium-sized stream CatBk appears particularly challenging, especially on sunny days when direct solar radiation makes a large contribution to the heat budget of streams. For example, on sunny days when hourly solar radiation exceeded 750 W m À2 at the open site, solar radiation at CatBk ranged between 105 W m À2 and 732 W m À2 , a sevenfold difference in abovestream values (Figure 3d). Moreover, determining a transmission coefficient representative of a stream reach would require numerous above-stream solar radiation measurements to capture the influence of sensor location (left, middle or right), canopy closure and stream orientation. From a modelling perspective, these medium-sized streams will require more computationally intensive approaches to simulate the solar radiation heat flux. For example, LiDAR (Loicq et al. 2018) and drone-based (Dugdale, Malcolm, and Hannah 2019) data have been used in recent years to characterize riparian vegetation in stream shading models. As these new modelling tools are being developed, careful thought will need to be put into the design of proper validation strategies.

Conclusion
This research used a direct measurement approach to describe the spatial and temporal variability in the proportion of incoming solar radiation reaching streams in a forested catchment. While various studies have modelled solar radiation reaching streams in forested catchments by simulating the solar position and taking into account the stream orientation and riparian vegetation geometry, very few have validated results with field measurements. By deploying abovestream microclimate stations, we thus gathered a dataset that can be used to better assess the performance of solar radiation models, and thus improve water temperature models.
We observed little hourly variability in the transmission coefficients of Trib and LSWM thus suggesting that, when using a direct measurement approach, a single pyranometer deployed mid-stream is sufficient to assess solar radiation reaching a headwater stream (Trib) or a wide river (LSWM) in a forested catchment. We also showed the importance of considering the seasonal variability of transmission coefficients for small headwater streams in deciduous forest catchments, as solar radiation transmission can decrease fourfold between the spring and summer.
We observed large hourly variability in solar radiation reaching the medium-sized stream (CatBk) as the transmission coefficient almost double between the morning when the stream was shaded and the afternoon as the stream received direct sunlight. Results suggest that daily modeling of stream temperatures will not be affected by this temporal variability; however, hourly stream temperature modeling need to consider such variation of solar radiation input. We also showed that there was considerable reachscale variability in transmission coefficients, even in a stream like CatBk with relatively undisturbed and mature riparian vegetation.
Significant advancements have been made over the past decades on stream temperature dynamics and on the overall understanding of stream heat fluxes (e.g. streambed heat fluxes, latent heat exchanges). However, solar radiation remains the largest contributing heat flux to streams during the summer and process-based stream models need to account for the spatial and temporal variations of this heat flux within stream environments. As shown by the present study, this is particularly important for medium-sized streams with intermediate canopy closure as well as for the implementation of stream temperature models at an hourly time scale. Given its central role in the heat budget of streams, rigorous monitoring of above-stream radiation is needed. While collecting abovestream solar radiation data that captures temporal and spatial variability is a demanding task, such dataset remains crucial to guide model parameterization and validation efforts.